A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions

TL;DR

This paper introduces new classes of multivariate and matrix Gamma distributions that are infinitely divisible, with applications to positive semi-definite matrices, and explores their properties and connections to Lévy processes and Wishart distributions.

Contribution

It develops a comprehensive framework for cone-valued Gamma distributions, characterizes generalised Gamma convolutions via Itô-Wiener integrals, and presents a novel infinitely divisible matrix Gamma distribution.

Findings

Introduction of a new infinitely divisible matrix Gamma distribution

Characterization of cone-valued Gamma convolutions using Lévy processes

Highlighting the relation between Lévy measure moments and Wishart distribution

Abstract

Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an It\^o-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a L\'evy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is introduced. It has properties which make it appealing for modelling under an infinite divisibility framework. An interesting relation of the moments of the L\'evy measure and the Wishart distribution is highlighted which we suppose to be important when considering the limiting distribution of the eigenvalues .